Theorems · Theorem · commutative algebra
squarefree_iff_emultiplicity_le_one
∀ {R : Type u_1} [inst : CommMonoid R] (r : R), Squarefree r ↔ ∀ (x : R), emultiplicity x r ≤ 1 ∨ IsUnit x- Defined in
- Mathlib.Algebra.Squarefree.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- ENatstatement and proof · cited by 4,985
- Nat.cast_oneproof · cited by 2,501
- CommMonoidstatement and proof · cited by 2,264
- IsUnitstatement and proof · cited by 1,602
- not_leproof · cited by 328
- sqproof · cited by 280
- emultiplicitystatement and proof · cited by 156
- Squarefreestatement · cited by 112
- one_add_one_eq_twoproof · cited by 65
- pow_dvd_iff_le_emultiplicityproof · cited by 11
- Order.add_one_le_iff_of_not_isMaxproof · cited by 3
Cited by4
Results whose statement or proof uses this declaration.
- Polynomial.Separable.squarefreeproof · cited by 6
- UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactorsproof · cited by 3
- Squarefree.natFactorization_le_oneproof · cited by 3
- IsPrimitiveRoot.minpoly_eq_powproof · cited by 1